A Stochastic Differential Game in the Closed-Loop Supply Chain with Third-Party Collecting and Fairness Concerns

This paper investigates the optimal return control problem in a closed-loop supply chain consisted of one manufacturer, one retailer, and one third-party collector, in the presence of stochastic return disturbance and fairness concern of followers. We formulate the stochastic differential game-theoretic models and resolve the feedback Stackelberg equilibriums without and with fairness concern. We also derive the evolutionary paths of the stochastic return rate and the value functions of the supply chain members under the optimal control strategies. We find that the feedback equilibrium exists only under a specific condition, and the expectation and variance of the return rate both approach the stable state for a specific closed-loop supply chain system. We further discussed the impact of fairness concerns on the supply chain system. The manufacturer would shift profit to the retailer by lowering the wholesale price, and the stable expected return rate will be lower in the supply chain with fairness concerns, as the third party will have less incentive to collect used products, considering unfairness. The manufacturer should set a higher transfer subsidy to incentivize the third party to collect when the third party is concerned with fairness.